Is the union a basis of that extension?

Question

Suppose that $H$ is a trancendental basis of the extension $A/F$ and $K$ is a trancendental basis of the extension $B/F$.

So, $H$ is the maximal among all the subsets of $A$ that are $F$-algebraic independent and $K$ is the maximal among all the subsets of $B$ that are $F$-algebraic independent.

Does it hold that $H\cup K$ is a trancendental basis of the extension $AB/F$ ?

Answer 1

Not necessarily. Consider $A=F(X^2)\subseteq B=F(X,Y),\ H=\{X^2\},\ K=\{X,Y\}$. When $A\subseteq B$.